# [R] glm.nb versus glm estimation of theta.

Achim Zeileis Achim.Zeileis at wu-wien.ac.at
Fri Aug 14 01:35:57 CEST 2009

```On Thu, 13 Aug 2009, hesicaia wrote:

>
> Hello,
>
> I have a question regarding estimation of the dispersion parameter (theta)
> for generalized linear models with the negative binomial error structure. As

The theta is different from the dispersion. In the usual GLM notation:
E[y] = mu
VAR[y] = dispersion * V(mu)

The function V() depends on the family and is
Poisson: V(mu) = mu
NB:      V(mu) = mu + 1/theta * mu^2

For both models, dispersion is known to be 1 (from the likelihood).
However, a quasi-Poisson approach can be adopted where dispersion is
estimated (but does not correspond to a specific likelihood).

Thus, dispersion and theta are really different things although both of
them can be used to capture overdispersion.

> I understand, there are two main methods to fit glm's using the nb error
> structure in R: glm.nb() or glm() with the negative.binomial(theta) family.
> Both functions are implemented through the MASS library. Fitting the model
> using these two functions to the same data produces much different results
> for me in terms of estimated theta and the coefficients, and I am not sure
> why.

...because you tell them to do different things.

> the following model:
> mod<-glm.nb(count ~ year + season + time + depth,
> estimates theta as 0.0109

This approach estimates theta (= 0.0109) and assumes that dispersion is
known to be 1. The underlying estimated variance function is:
VAR[y] = mu + 91.74312 * mu^2

> while the following model:
> mod2<-glm(count ~ year + season + time + depth,
> will not accept 0.0109 as theta and instead estimates it as 1271 (these are
> fisheries catch data and so are very overdispersed).

This does not estimate theta at all but keeps it fixed (= 100). By
default, however, dispersion will be estimated by the summary() method,
presumably leading to the value of 1271 you report. The underlying
variance function would then be
VAR[y] = 1271 * mu + 12.71 * mu^2

> Fitting a quasipoisson model also yields a large dispersion parameter
> (1300). The models also produce different coefficients and P-values, which
> is disconcerting.

This implies yet another variance function, namely
VAR[y] = 1300 * mu

If you want to get essentially the same result as
summary(mod)
from using glm+negative.binomial you can do
mod0 <- glm(count ~ year + season + time + depth, data = dat,
family = negative.binomial(theta = mod\$theta),
control = glm.control(maxit = 100))
summary(mod0, dispersion = 1)
(Note that link = "log" is not needed.)

> What am I doing wrong here? I've read through the help sections
> (?negative.binomial,?glm.nb, and ?glm) but did not find any answers.

I guess the authors of the "MASS" package would say that the software
accompanies a book which should be consulted...and they would be right.
Reading the corresponding sections in MASS (the book) will surely be
hurt either. Finally, some extended modeling (including excess zeros) is
available in
http://www.jstatsoft.org/v27/i08/
(Apologies for advertising this twice on the same day.)

hth,
Z

> Any help and/or input is greatly appreciated!
> Daniel.
> --
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